Null energy conditions in quantum field theory

TL;DR

This paper demonstrates that in four-dimensional Minkowski space, quantum states can violate null energy conditions when averaged along null geodesics, contrasting with two-dimensional cases and affecting singularity theorem considerations.

Contribution

It provides an explicit construction showing the nonexistence of null quantum inequalities in four-dimensional Minkowski space, highlighting differences from lower-dimensional cases.

Findings

Weighted averages of stress-energy along null geodesics are unbounded from below.

Null quantum inequalities do not exist in four-dimensional Minkowski space.

Null energy condition holds on average over timelike worldlines in any globally hyperbolic spacetime.

Abstract

For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems.